By expanding \(\psi (\mathbf {r})\)  in terms of eigenfunctions of the operator corresponding to \(J\)  one can project the Schrödinger equation (REF ) to definite \(J\)  (for details see Refs. [1]}, [2]}). This leads to a set of five coupled ordinary differential equations in the radial coordinate \(r\) ,
\(\nonumber & & \hspace{-19.91684pt} \bigg (\frac{1}{2}\mu ^{-1}\,\partial _r^2 + \frac{1}{2r^2}L_{J}^2 + V_{J}(r) + \\\nonumber & & \hspace{19.20569pt} +\left.\left(\begin{array}{ccccc}E_{\text{threshold}} & 0 & 0 & 0 & 0 \\0 & 2m_M & 0 & 0 & 0 \\0 & 0 & 2m_M & 0 & 0 \\0 & 0 & 0 & 2m_{M_s} & 0 \\0 & 0 & 0 & 0 & 2m_{M_s} \\\end{array}\right)- E\;{1}_{5\times 5} \right)\left( \begin{array}{c} u_{J}(r) \\ \chi _{\bar{M}M,J-1 \rightarrow J}(r) \\ \chi _{\bar{M}M,J+1 \rightarrow J}(r) \\ \chi _{\bar{M_s}M_s,J-1 \rightarrow J}(r) \\ \chi _{\bar{M_s}M_s,J+1 \rightarrow J}(r) \end{array}\right) = \\\nonumber & & = \left( \begin{array}{c} V_{\text{mix}}(r) \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right) \bigg ( \alpha _{\bar{M}M,1} \; {J\over 2 J+1} \; r \; j_{J-1}(kr) + \alpha _{\bar{M}M,2} \; {J+1 \over 2J+1} \; r \; j_{J+1}(kr) + \\& & \hspace{19.20569pt} + \alpha _{\bar{M_s}M_s,1} \; {J\over 2 J+1} \frac{r \; j_{J-1}(k_sr)}{\sqrt{2}} + \alpha _{\bar{M_s}M_s,2} \; {J+1 \over 2J+1} \frac{r \; j_{J+1}(k_sr)}{\sqrt{2}}\bigg ) ,\) 
